1. **State the problem:** Find all positive integers $x, y, z$ such that $$x^2 + y^2 = z^2$$ and $$x + y + z = 1000.$$ This means we are looking for a Pythagorean triple $(x,y,z)$ whose sum is 1000. 2. **Formula and rules:** A Pythagorean triple satisfies the equation $x^2 + y^2 = z^2$. We also have the sum condition $x + y + z = 1000$. 3. **Approach:** We can express $z$ as $z = 1000 - x - y$ and substitute into the Pythagorean equation: $$x^2 + y^2 = (1000 - x - y)^2.$$ Expanding the right side: $$x^2 + y^2 = 1000^2 - 2 \cdot 1000 (x + y) + (x + y)^2.$$ Simplify: $$x^2 + y^2 = 1,000,000 - 2000(x + y) + x^2 + 2xy + y^2.$$ 4. **Cancel $x^2$ and $y^2$ on both sides:** $$0 = 1,000,000 - 2000(x + y) + 2xy.$$ Rearranged: $$2xy = 2000(x + y) - 1,000,000.$$ Divide both sides by 2: $$xy = 1000(x + y) - 500,000.$$ 5. **Rewrite as:** $$xy - 1000x - 1000y = -500,000.$$ Add $1,000,000$ to both sides to complete the rectangle: $$xy - 1000x - 1000y + 1,000,000 = 500,000.$$ Factor left side as: $$(x - 1000)(y - 1000) = 500,000.$$ 6. **Find integer factor pairs of 500,000:** Since $x$ and $y$ are positive integers, $x - 1000$ and $y - 1000$ are integers whose product is 500,000. 7. **Check factor pairs:** Try $x - 1000 = 200$, then $y - 1000 = \frac{500,000}{200} = 2500$. So $x = 1200$, $y = 3500$ which is invalid since $x + y + z$ would exceed 1000. Try $x - 1000 = -200$, then $y - 1000 = \frac{500,000}{-200} = -2500$ which is negative. Try $x - 1000 = -375$, then $y - 1000 = \frac{500,000}{-375} = -1333.33$ not integer. Try $x - 1000 = -800$, then $y - 1000 = \frac{500,000}{-800} = -625$. So $x = 200$, $y = 375$. 8. **Calculate $z$:** $$z = 1000 - x - y = 1000 - 200 - 375 = 425.$$ 9. **Verify Pythagorean triple:** $$200^2 + 375^2 = 40,000 + 140,625 = 180,625,$$ $$425^2 = 180,625.$$ So the triple $(200, 375, 425)$ satisfies both conditions. 10. **Answer:** The unique Pythagorean triple with sum 1000 is: $$(x, y, z) = (200, 375, 425)$$ or $$(375, 200, 425).$$